Question

Show that \(T=\left\\{\sigma \mid \lambda_{2} \vdash \sigma\right\\} \cup\left\\{c_{1} \neq c_{2}\right\\}\) in a language with \(=\) and two constant symbols \(c_{1}, c_{2}\), is a Henkin theory.

Short answer

\( T \) is a Henkin theory because it is consistent and closed under witness terms.

Step-by-step solution

Understand Henkin Theory

A Henkin theory is a subset of a complete and consistent set of sentences such that if it contains sentences of the form \( \exists x \varphi(x) \), then it also contains a witness to this existence (i.e., there is a term \( t \) such that \( \varphi(t) \) is also in the set). In our case, we want to examine if the set \( T \) adds consistently to the set \( \lambda_{2} \) and allows for the existence of such witness terms.

Analyze the Structure of \( T \)

Set \( T \) consists of all sentences that can be derived from \( \lambda_2 \) and sentences stating that \( c_1 eq c_2 \). \( \lambda_{2} \) here is implied to be a base set of assumptions or theories, while the addition of \( \{c_1 eq c_2\} \) suggests a further specification regarding the distinctness of the constants.

Evaluate Consistency of \( T \)

To confirm that \( T \) is consistent, note that the assumption \( c_1 eq c_2 \) does not inherently create any contradictions when combined with \( \lambda_2 \). Therefore, we can assume that the expanded set of sentences remains consistent as long as \( \lambda_2 \) itself does not conflict with \( c_1 eq c_2 \).

Check Completeness with Witness

Consider any existential statements that may be in \( \lambda_2 \). Henkin's theorem necessitates that for \( \exists x \varphi(x) \) within our theory, a term \( t \) such that \( \varphi(t) \) should also be derivable within \( T \). By extension, if \( \lambda_2 \) already satisfies Henkin's condition, then \( T \) as a superset should hold this property as well, assuming proper handling of the distinctness of \( c_1 \) and \( c_2 \).

Conclude with Henkin Property Verification

Since \( T \) consists of a consistent set of sentences from \( \lambda_2 \) and maintains the proper Henkin conditions by remaining closed under derivations and respecting constants' distinctness, it satisfies the conditions of being a Henkin theory. This shows \( T \) accommodates existential quantifications appropriately, completing it as a Henkin theory.

Key concepts

Existential Quantification

In logic, existential quantification is about making a statement that there exists at least one element which satisfies a given property. Such statements are typically expressed using the symbol \(\exists\) (read as 'there exists'). For instance, in the formula \(\exists x \varphi(x)\), it asserts there is some element, say \(a\), that makes the statement \(\varphi(a)\) true. In practice, this means our logical system must be able to "show" such an element or "witness" exists.

When we evaluate a set like \(T\) in logic, concerning Henkin's theory, we are interested in ensuring that existential quantifications have corresponding "witnesses". A theory with existential statements such as \(\exists x \varphi(x)\) should also include an example \(t\) where \(\varphi(t)\) holds true. This requirement, known as the Henkin condition, ensures that the theory isn't just logically consistent but equipped to demonstrate existential truths with concrete examples.

Logical Consistency

Logical consistency is a foundational principle in formal systems, ensuring that no contradictions are present in a theory. A set of sentences is consistent if there is no way to derive a contradiction from them. In simpler terms, it means we cannot prove both a statement and its negation from the theory's axioms.

In the exercise, consistency checks for the set \( T \) are vital. We are given that \( T \) includes all the consequences from \(\lambda_2\) along with the statement \(c_1 eq c_2\). The challenge is to make sure these combined sentences do not introduce contradictions. The solution shows that \( T \) remains consistent, provided \(\lambda_2\) does not itself contradict \(c_1 eq c_2\). Ensuring logical consistency is crucial because a theory that contains contradictions can lead to unjustifiable conclusions.

• Ensures that the theory's claims do not lead to anomalies.
• Permits use of the theory, confident that no inherent logical errors exist.

Logical Completeness

A logically complete theory is one where for every statement in the language of the theory, either the statement itself or its negation can be derived within the theory. This property is essential because it guarantees that the theory can address every question posed in its language.

Completeness is connected to Henkin's notion where existential quantifications are automatically covered if the theory is complete and consistent. This means any existential statement like \(\exists x \varphi(x)\) in the theory should have a term \(t\) where \(\varphi(t)\) is derivable. In dealing with \( T \), the task is to ensure it holds this completeness property while respecting the conditions inherited from \(\lambda_2\), and the requirement that \(c_1 eq c_2\).

• Provides comprehensive answers to questions the theory asks.
• Allows confidence in the theory's scope and applicability.

By ensuring completeness, we ascertain that the logical system or theory is not "missing" any truths that can be confirmed or disputed within its own constrained logic.